1. Pick a region for your X-COM base. Kill off the four sections in the opening mission. Get a research project going when you return to base. Beam weaponry is a good starting point so that you can arm your soldiers with more powerful laser guns.
2. The next mission is an abduction. Choose whichever city has the highest levels of panic in order to decrease the unrest and level your map out. Despite that obvious benefit, you should also base your decision on the rewards. Go for engineers or scientists at this point. They'll enable you to create new toys at this early stage in the game.
3. As the campaign progresses at this point you'll be able to capture a live specimen for Dr. Vahlen. You don't have to do this, but the longer you sit on PRIORITY missions, the more powerful the aliens become. You can't sit and grind forever. From this point on, as soon as you see a new alien, it's in your best interest to capture it live and benefit from the research.
Of course, you'll also have to have the facilities in place. Get to building!
4. Satellites. Around the same time the ARC Thrower is developed you'll be able to swing a few satellites and uplink facilities. Be sure to build the Uplink before you spend money on Satellites. It's kind of a waste if they sit in the warehouse for very long.
Satellites will let you catch UFOs and take them down with your interceptors. Don't forget that launching a satellite decreases the panic in that area as well.
5. The conflict quickly escalates until you find yourself up against spooky looking aliens that materialize out of thin air. Stunning one of these nets you an Outsider Shard. This is the next big piece of research. Just keep studying what you bring in and developing your base to accommodate your forces and the responsibilities you have around the world. Once you've done your research, you'll obtain the Skeleton Key which unlocks a huge Alien base.
Get in there and f*** s*** up. I suggest you don't hesitate. Again, the aliens scale to meet you, so sitting around on your hands won't help your situation in the base.
6. With the Hyperwave Relay comes more research and the option to build your own. Doing so will allow you to scan for a new and more powerful UFO. The campaign begins to escalate at this point, both in difficulty and on its way to the conclusion. Don't hesitate on these new research opportunities as doing so will only allow the aliens to increase their own offensive capabilities.
7. Even after you've taken down the Overseer UFO and it's mind-controlling Ethereal, an EVEN BIGGER UFO will show up. It should be pretty clear that you'll need to do your research, update your facilities, and gear up for another assault.
8. Now build a Gallop chamber to trigger the final chapter of the game. Once the chamber has been built in your secret underground facility, you'll have to have a developed Psionic soldier to activate it. If you're having trouble, give a soldier the PSI Armor.
9. With the chamber activated, you're locked into the final mission. No scanning, no building, nothing more. Make sure you're ready at this point because the only thing you can do is build items that take 0 time to develop. Try to have Mind shields for everyone if you don't have soldiers with very high Will. Bring a balanced team to the Temple Ship.
There you'll find what amounts to a long corridor packed with baddies. Take it slow, use Overwatch a lot and focus fire on specific threats. You can do it!
"We'll be watching, Commander."